Why the Fermi level determines the conductivity of the material? Considering electronic band structure in solid state Physics, the Fermi level is defined as the chemical potential appearing at the Fermi-Dirac distribution
$$f(\epsilon)=\dfrac{1}{\exp[(\epsilon-\mu)/k_BT]+1}.$$
In that sense, it is just one specific value of energy that we chose to give a name to it.
Now, the Fermi level is quite important because of the following points, according to Wikipedia's article:


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*In an insulator, $\mu$ lies within a large band gap, far away from any states that are able to carry current.

*In a metal, semimetal or degenerate semiconductor, $\mu$ lies within a delocalized band. A large number of states nearby $\mu$ are thermally active and readily carry current.

*In an intrinsic or lightly doped semiconductor, $\mu$ is close enough to a band edge that there are a dilute number of thermally excited carriers residing near that band edge.
In other words, it seems that the conductivity properties of the material are determined by $\mu$.
But why is that? Why $\mu$ has all these properties?
How can we actually find out these properties of the Fermi level? How can we find out that the Fermi level determines the conductivity according to these points?
 A: As you have already indicated, the Fermi level $\mu$ determines the Fermi-Dirac energy distribution of electrons (and holes) in the solid. Thus in a semiconductor at a given temperature, if the Fermi level is close to the conduction band (brought about by doping with donor atoms), a large electron concentration appears in the conduction band together with a low hole concentration in the valence band. Conversely, if the Fermi level is close to the valence band (by acceptor doping), you get a large hole concentration there and a low electron concentration in the conduction band. In an intrinsic semiconductor (no doping) the fermi level is close to the midgap energy and you have equal electron and hole concentrations in the conduction and valence band, respectively. Similarly, in an insulator with a large bandgap, the Fermi level is in the middle of the band gap and you have very low electron and hole concentrations. In metals the Fermi level lies in a partly filled band and it can be shown that conduction is due to the electrons close in energy to the Fermi level. Thus the conduction properties are determined by the energetic position of the Fermi level with respect to the conduction and valence bands or by the position in an energetically allowed band of a metal or degenerate semiconductor.
Note: See my above comment on temperature dependence of semiconductor carrier concentration. 
